Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in ${\mathbb{CP}}^2$ as $a+b+c\to \infty$

Abstract

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in ${\mathbb{CP}}^2$ associated to Markov triples $(a,b,c)$ described in \cite{Vi14}. We first prove that the Gromov capacity of the complement ${\mathbb{CP}}^2\setminus T_{a,b,c}$ is greater than or equal to $\frac{1}{3}$ of the area of the complex line for all Markov triple $(a,b,c)$. We then prove that there is a representative of the family $\{T_{a,b,c}\}$ whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in ${\mathbb{CP}}^2$.

24 pages, 7 figures;v2) typos corrected, English improved, More references added;v3) 32 pages, 16 figures, the third author newly added, previous main theorem improved, density question resolved and many more results on relative ball packings added

References in corpus (1)

• Symplectic Rational Blow-up